series

[odd wave]

im a bit confused on how to proceed with 14 to 16 like
i think tis geometric series but im a bit confused idk

[loud seal]

is 14 a nul sequence?

[odd wave]

whats that

[loud seal]

check if the sequence converges to 0 first

necessary condition for convergence of series

[odd wave]

like do the test?

[loud seal]

yes

[odd wave]

ok

[loud seal]

k(k+2)/(k+3)² does this go to 0?

[odd wave]

it does to 1

goes*

means its convergent right

[loud seal]

this means the series doesn't converge

for a series to converge it's necessary that its sequence converges to 0

[odd wave]

ooohh yeah yeah

ok so its divergent

ok thank u so much

[loud seal]

Same wirh 15

[odd wave]

yes 15 i managed

now im doing 16

[loud seal]

16 years in prison

[odd wave]

the test is going nowhere since i need to do hr and it keeps giving infinity over inifinity

[loud seal]

yea

3^n is faster than 2^n

so it's also not a null sequence

[odd wave]

what do these both sentence mean

[loud seal]

The limit

Of

(1+3^n)/2^n is not 0

Because the infinity in the nunerator is faster and thus bigger than in the denominator

So the seq. diverges

and the series does too

[odd wave]

sorry i rlly dont get what that means

how do u know one is faster like what is faster

it reaches a certain numebr faster for example?

[loud seal]

3^n is faster than 2^n

because 3 is bigger base than 2

[odd wave]

ok i see

but then how do u know its not 0

i tried evaluating the limit

i kept getting infinity over infininty

[loud seal]

factor 3^n

[odd wave]

how

[loud seal]

Try

[odd wave]

333... n times...?

[loud seal]

No

1/3^n + 1

[odd wave]

hoowww

[loud seal]

now evaluate the limit it's obv

[odd wave]

oooooooohhhhhhhhhhhhhhhhhh

its inifinity right

wow ok i see then

thank u so much

[loud seal]

yooo

[tired heron]

𝔸dωn𝓲²s

[loud seal]

This is simply how factorization works

[odd wave]

ok thank u sooo much!!!!!!

[loud seal]

when you pull something out, you have to divide every term by it

1+3^n i pull out 3^n(...) i need to divide by 3^n in (...) to make up for it

[odd wave]

ok perfect i see thankss

im also struggling with this one could u help if you dont mind

and for 23 and 24 i would have said both diverge because there is not like a fixed continuity but 24 converges and im confused why and idk if my reasoning is good

[loud seal]

both are geometric series, so split it

and use the formula

also you need an index shift

[tired heron]

𝔸dωn𝓲²s

[loud seal]

look like 1/3^n which is a geometric series

and it converges